What is the intuitive meaning of uncorrelated?

Question

I was going through the derivation of the Kalman filter and it mentions that since noise (v) is uncorrelated to the state (x) and the state estimate (xbar), the following quantity is zero:

E((x - xbar) * v); where E is the expected value

I would like to know the meaning of uncorrelated intuitively as well? Thank you.

Answer 1

"Uncorrelated" means that as one of the two variables increases, on average the other one does not increase or decrease.

That does not mean they are independent: for example, suppose the two variables are the $x$- and $y$-coordinates of the point $(x,x^2)$ where $x\in\{-1,0,1\}$, each with equal probability. If you know $x$, that determines $y$, so they are not independent, but they are uncorrelated.

The images below are scatterplots with respective correlations $0.99$, $0.95$, $0.9$, $0.7$, $0.5$, $0.3$, $0$, $-0.3$, $-0.5$, $-0.7$, $-0.9$, $-0.95$, $-0.99$. The one case in which the correlation is $0$ is the case in which the two variables are uncorrelated. If the correlation were exactly $1$ or exactly $-1$, then all of the points would lie on a single straight line.

correlation $=0.99$:

$=0.99$" />

correlation $=0.95$:

$=0.95$" />

correlation $=0.9$:

correlation $=0.7$:

$=0.7$" />

correlation $=0.5$:

correlation $=0.3$:

$=0.3$" />

correlation $=0$:

$=0$" />

correlation $=-0.3$:

$=-0.3$" />

correlation $=-0.5$:

$=-0.5$" />

correlation $=-0.7$:

$=-0.7$" />

correlation $=-0.9$:

$=-0.9$" />

correlation $=-0.95$:

$=-0.95$" />

correlation $=-0.99$:

$=-0.99$" />

Answer 2

My interpretation of "uncorrelated" is that knowing information about one variable does not supply us with any information about the other variable.

(My apologies for not have any plots.)